Question

question 3 of 25
which of the following is a result of shifting a circle with equation $(x + 3)^2+(y - 2)^2 = 36$ left 3 units?
a. the x - coordinate of the center of the circle decreases by 3.
b. both the x - and y - coordinates of the center of the circle decrease by 3.
c. the x - coordinate of the center of the circle increases by 3.
d. the y - coordinate of the center of the circle increases by 3.

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius. For the given circle \((x + 3)^{2}+(y - 2)^{2}=36\), we can rewrite it as \((x-(- 3))^{2}+(y - 2)^{2}=6^{2}\). So the center of the original circle is \((-3,2)\) and the radius \(r = 6\).

Step2: Apply the horizontal shift rule

When a circle is shifted left \(a\) units, the transformation rule for the \(x\) - coordinate of the center is \(h\to h - a\) (while the \(y\) - coordinate \(k\) remains the same). Here \(a = 3\). After shifting the circle \((x-(-3))^{2}+(y - 2)^{2}=6^{2}\) left 3 units, the new \(x\) - coordinate of the center is \(-3-3=-6\), and the \(y\) - coordinate remains \(2\). The change in the \(x\) - coordinate is \(\Delta x=-6-(-3)=-3\), which means the \(x\) - coordinate of the center decreases by 3. The \(y\) - coordinate of the center does not change.

Answer:

A. The x - coordinate of the center of the circle decreases by 3.